Incomplete elliptic integral of the first kind: Series representations (formula 08.05.06.0055)

EllipticF

$Mathematica Notation$

Elliptic Integrals

EllipticF[z,m]

Series representations

Generalized power series

Expansions at generic point m==m₀

For the function itself

http://functions.wolfram.com/08.05.06.0055.01

Input Form

EllipticF[z, m] \[Proportional] EllipticF[z, Subscript[m, 0]] + (1/4) ((2 EllipticE[z, Subscript[m, 0]])/(Subscript[m, 0] (1 - Subscript[m, 0])) - (2 EllipticF[z, Subscript[m, 0]])/ Subscript[m, 0] - Sin[2 z]/((1 - Subscript[m, 0]) Sqrt[1 - Subscript[m, 0] Sin[z]^2])) (m - Subscript[m, 0]) + (1/(16 (-1 + Subscript[m, 0])^2 Subscript[m, 0]^2 (1 - Sin[z]^2 Subscript[m, 0])^(3/2))) (2 (1 - Sin[z]^2 Subscript[m, 0])^(3/2) (2 EllipticE[z, Subscript[m, 0]] (-1 + 2 Subscript[m, 0]) + EllipticF[z, Subscript[m, 0]] (-1 + Subscript[m, 0]) (-2 + 3 Subscript[m, 0])) + Sin[2 z] Subscript[m, 0] (1 - (3 + 2 Sin[z]^2) Subscript[m, 0] + 4 Subscript[m, 0]^2 Sin[z]^2)) (m - Subscript[m, 0])^2 + O[(m - Subscript[m, 0])^3]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["EllipticF", "[", RowBox[List["z", ",", SubscriptBox["m", "0"]]], "]"]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["EllipticE", "[", RowBox[List["z", ",", SubscriptBox["m", "0"]]], "]"]]]], RowBox[List[SubscriptBox["m", "0"], RowBox[List["(", RowBox[List["1", "-", SubscriptBox["m", "0"]]], ")"]]]]], "-", FractionBox[RowBox[List["2", " ", RowBox[List["EllipticF", "[", RowBox[List["z", ",", SubscriptBox["m", "0"]]], "]"]]]], SubscriptBox["m", "0"]], "-", FractionBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "z"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["m", "0"]]], ")"]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SubscriptBox["m", "0"], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]]]]]], ")"]], RowBox[List["(", RowBox[List["m", "-", SubscriptBox["m", "0"]]], ")"]]]], " ", "+", RowBox[List[FractionBox["1", RowBox[List["16", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["m", "0"]]], ")"]], "2"], " ", SubsuperscriptBox["m", "0", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List[SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], " ", SubscriptBox["m", "0"]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List[SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], " ", SubscriptBox["m", "0"]]]]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["EllipticE", "[", RowBox[List["z", ",", SubscriptBox["m", "0"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", SubscriptBox["m", "0"]]]]], ")"]]]], "+", RowBox[List[RowBox[List["EllipticF", "[", RowBox[List["z", ",", SubscriptBox["m", "0"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["m", "0"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["3", " ", SubscriptBox["m", "0"]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["Sin", "[", RowBox[List["2", " ", "z"]], "]"]], " ", SubscriptBox["m", "0"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], ")"]], " ", SubscriptBox["m", "0"]]], "+", RowBox[List["4", " ", SubsuperscriptBox["m", "0", "2"], SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SubscriptBox["m", "0"]]], ")"]], "2"]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SubscriptBox["m", "0"]]], ")"]], "3"], "]"]]]]]]]]

MathML Form

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</mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msubsup> <mi> m </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <ci> EllipticF </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <ci> EllipticF </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticE </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticF </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> EllipticE </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <ci> EllipticF </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> -2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02